Question 1 - A-Level Maths

AQA S2 2007 January — Question 1 5 marks

Exam Board	AQA
Module	S2 (Statistics 2)
Year	2007
Session	January
Marks	5
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	T-tests (unknown variance)
Type	Single sample confidence interval t-distribution
Difficulty	Moderate -0.3 This is a straightforward application of the t-distribution confidence interval formula with given data. Students need to calculate sample mean and standard deviation, then apply the standard formula with t₇ critical value. While it requires careful arithmetic and knowledge of the procedure, it's a routine textbook exercise with no conceptual challenges or problem-solving required beyond direct application.
Spec	5.05d Confidence intervals: using normal distribution

1 Alan's journey time, in minutes, to travel home from work each day is known to be normally distributed with mean $\mu$. Alan records his journey time, in minutes, on a random sample of 8 days as being $$\begin{array} { l l l l l l l l } 36 & 38 & 39 & 40 & 50 & 35 & 36 & 42 \end{array}$$ Construct a $95 \%$ confidence interval for $\mu$.

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Question 1:

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
$\bar{x} = 39.5$, $s = 4.84$ ($s^2 = 23.4$)	B1B1	$\sigma = 4.53$ ($\sigma^2 = 20.5$)
$t_{\text{crit}} = 2.365$	B1
$= \bar{x} \pm t_{\text{crit}} \times \dfrac{s}{\sqrt{n}}$	M1	$39.5 \pm 2.365 \times \dfrac{4.53}{\sqrt{7}}$
$= 39.5 \pm 2.365 \times \dfrac{4.84}{\sqrt{8}}$
$= 39.5 \pm 4.05$
$= (35.5, 43.5)$	A1$\checkmark$	Total: 5

## Question 1:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\bar{x} = 39.5$, $s = 4.84$ ($s^2 = 23.4$) | B1B1 | $\sigma = 4.53$ ($\sigma^2 = 20.5$) |
| $t_{\text{crit}} = 2.365$ | B1 | |
| $= \bar{x} \pm t_{\text{crit}} \times \dfrac{s}{\sqrt{n}}$ | M1 | $39.5 \pm 2.365 \times \dfrac{4.53}{\sqrt{7}}$ |
| $= 39.5 \pm 2.365 \times \dfrac{4.84}{\sqrt{8}}$ | | |
| $= 39.5 \pm 4.05$ | | |
| $= (35.5, 43.5)$ | A1$\checkmark$ | **Total: 5** |

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1 Alan's journey time, in minutes, to travel home from work each day is known to be normally distributed with mean $\mu$.

Alan records his journey time, in minutes, on a random sample of 8 days as being

$$\begin{array} { l l l l l l l l } 
36 & 38 & 39 & 40 & 50 & 35 & 36 & 42
\end{array}$$

Construct a $95 \%$ confidence interval for $\mu$.

\hfill \mbox{\textit{AQA S2 2007 Q1 [5]}}

This paper (8 questions)

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Q1 5 Q2 13 Q3 8 Q4 9 Q5 8 Q6 14 Q7 10 Q8 8

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
\(\bar{x} = 39.5\), \(s = 4.84\) (\(s^2 = 23.4\))	B1B1	\(\sigma = 4.53\) (\(\sigma^2 = 20.5\))
\(t_{\text{crit}} = 2.365\)	B1
\(= \bar{x} \pm t_{\text{crit}} \times \dfrac{s}{\sqrt{n}}\)	M1	\(39.5 \pm 2.365 \times \dfrac{4.53}{\sqrt{7}}\)
\(= 39.5 \pm 2.365 \times \dfrac{4.84}{\sqrt{8}}\)
\(= 39.5 \pm 4.05\)
\(= (35.5, 43.5)\)	A1\(\checkmark\)	Total: 5